These methods consist of more or less elaborate approximations of the Boltzmann equation. The most widely used approximation is the multigroup diffusion theory which we outline here, as an example. The different groups correspond to energy bands Ei < E < Ei +1. The set of multigroup equations reads

DiA’i(r)- (r) + X! Sr, j!i’j(r) + Xi^2v^f, j’j(r) (5-8)

j j

where i( j) denotes the i( j)th group. Xr, j^ i is the cross-section for a jump from group j to group i. St i = Sa i + j Xr, j i is the cross-section for removing neutrons from group i. Xf, j is the fission cross-section in group j.

Xi is the fraction of the fission neutrons which have energies within group i. The cross-sections should be computed as averages over the group energy domain by

(E)'(E) dE

which means that equation (5.8) is, in fact, a set of complicated integro — differential equations. In particular, in the resonance regions, the flux has a complicated structure due to its depletion at energies in the vicinity of a resonance energy. Thus, approximations are made on the calculation of the group cross-sections. In particular, in heterogeneous reactors the cross­sections for the cells are first computed, with a large number of groups, with simplifying assumptions on the shape of the flux, and, possibly, correc­tion factors. In a second step the flux on the cell network is computed. In practice, experiments are needed to validate the group cross-sections for each type of reactor.